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First-order perturbation of a simple eigenvalue

If

A (t) = A + tE

and

λ (t)

is a differentiable function defining a simple eigenvalue, then

λ^{'} (0) = \frac{u ^{T} E v}{u ^{T} v} .

The Cauchy-Schwarz inequality and the operator norm give

∣ λ^{'} (0) ∣ \leq ∥ E ∥_{2} s (λ) .

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